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# p2 - Problem Set 2 Physics 341 Due October 22 Some...

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Problem Set 2 Physics 341 Due October 22 Some abbreviations: S - Shankar. 1 . The Dirac delta-function has many nice properties. Show that δ ( ax ) = δ ( x ) | a | . Show that the limit δ ( x ) = lim a 0 1 π a x 2 + a 2 satisfies the properties of the Dirac delta-function. Namely, that δ ( x ) vanishes if x 6 = 0 and Z + -∞ δ ( x ) = 1 . 2 . Let’s continue with an exercise involving Gram-Schmidt. (i) Take the following three vectors in R 3 , | e 1 i = (1 , 1 , 1) , | e 2 i = (1 , 1 , 0) , | e 3 i = (1 , 0 , 1) . Construct an orthonormal basis using the Gram-Schmidt procedure starting with | e 1 i . (ii) Now let’s return to normalizable functions on the interval [ - 1 , 1]. Consider the inner product h f | g i = Z 1 - 1 f * ( x ) g ( x ) dx on this space. We can expand these functions in the basis { 1 , x, x 2 , x 3 , . . . } but this is not an orthonormal basis! We can build an orthonormal basis using Gram-Schmidt. For example, we can choose φ 0 = 1 2 , φ 1 = r 3 2 x, for the first two basis elements. Please construct φ 2 and φ 3 . These orthogonal polynomials are called Legendre polynomials.

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p2 - Problem Set 2 Physics 341 Due October 22 Some...

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